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G = C2×C108order 216 = 23·33

Abelian group of type [2,108]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C108, SmallGroup(216,9)

Series: Derived Chief Lower central Upper central

C1 — C2×C108
C1C3C9C18C54C108 — C2×C108
C1 — C2×C108
C1 — C2×C108

Generators and relations for C2×C108
 G = < a,b | a2=b108=1, ab=ba >


Smallest permutation representation of C2×C108
Regular action on 216 points
Generators in S216
(1 116)(2 117)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 146)(32 147)(33 148)(34 149)(35 150)(36 151)(37 152)(38 153)(39 154)(40 155)(41 156)(42 157)(43 158)(44 159)(45 160)(46 161)(47 162)(48 163)(49 164)(50 165)(51 166)(52 167)(53 168)(54 169)(55 170)(56 171)(57 172)(58 173)(59 174)(60 175)(61 176)(62 177)(63 178)(64 179)(65 180)(66 181)(67 182)(68 183)(69 184)(70 185)(71 186)(72 187)(73 188)(74 189)(75 190)(76 191)(77 192)(78 193)(79 194)(80 195)(81 196)(82 197)(83 198)(84 199)(85 200)(86 201)(87 202)(88 203)(89 204)(90 205)(91 206)(92 207)(93 208)(94 209)(95 210)(96 211)(97 212)(98 213)(99 214)(100 215)(101 216)(102 109)(103 110)(104 111)(105 112)(106 113)(107 114)(108 115)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)

G:=sub<Sym(216)| (1,116)(2,117)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,161)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,168)(54,169)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,183)(69,184)(70,185)(71,186)(72,187)(73,188)(74,189)(75,190)(76,191)(77,192)(78,193)(79,194)(80,195)(81,196)(82,197)(83,198)(84,199)(85,200)(86,201)(87,202)(88,203)(89,204)(90,205)(91,206)(92,207)(93,208)(94,209)(95,210)(96,211)(97,212)(98,213)(99,214)(100,215)(101,216)(102,109)(103,110)(104,111)(105,112)(106,113)(107,114)(108,115), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)>;

G:=Group( (1,116)(2,117)(3,118)(4,119)(5,120)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,148)(34,149)(35,150)(36,151)(37,152)(38,153)(39,154)(40,155)(41,156)(42,157)(43,158)(44,159)(45,160)(46,161)(47,162)(48,163)(49,164)(50,165)(51,166)(52,167)(53,168)(54,169)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,183)(69,184)(70,185)(71,186)(72,187)(73,188)(74,189)(75,190)(76,191)(77,192)(78,193)(79,194)(80,195)(81,196)(82,197)(83,198)(84,199)(85,200)(86,201)(87,202)(88,203)(89,204)(90,205)(91,206)(92,207)(93,208)(94,209)(95,210)(96,211)(97,212)(98,213)(99,214)(100,215)(101,216)(102,109)(103,110)(104,111)(105,112)(106,113)(107,114)(108,115), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216) );

G=PermutationGroup([(1,116),(2,117),(3,118),(4,119),(5,120),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,128),(14,129),(15,130),(16,131),(17,132),(18,133),(19,134),(20,135),(21,136),(22,137),(23,138),(24,139),(25,140),(26,141),(27,142),(28,143),(29,144),(30,145),(31,146),(32,147),(33,148),(34,149),(35,150),(36,151),(37,152),(38,153),(39,154),(40,155),(41,156),(42,157),(43,158),(44,159),(45,160),(46,161),(47,162),(48,163),(49,164),(50,165),(51,166),(52,167),(53,168),(54,169),(55,170),(56,171),(57,172),(58,173),(59,174),(60,175),(61,176),(62,177),(63,178),(64,179),(65,180),(66,181),(67,182),(68,183),(69,184),(70,185),(71,186),(72,187),(73,188),(74,189),(75,190),(76,191),(77,192),(78,193),(79,194),(80,195),(81,196),(82,197),(83,198),(84,199),(85,200),(86,201),(87,202),(88,203),(89,204),(90,205),(91,206),(92,207),(93,208),(94,209),(95,210),(96,211),(97,212),(98,213),(99,214),(100,215),(101,216),(102,109),(103,110),(104,111),(105,112),(106,113),(107,114),(108,115)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)])

C2×C108 is a maximal subgroup of   C4.Dic27  Dic27⋊C4  C4⋊Dic27  D54⋊C4  D1085C2

216 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F9A···9F12A···12H18A···18R27A···27R36A···36X54A···54BB108A···108BT
order12223344446···69···912···1218···1827···2736···3654···54108···108
size11111111111···11···11···11···11···11···11···11···1

216 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C6C6C9C12C18C18C27C36C54C54C108
kernelC2×C108C108C2×C54C2×C36C54C36C2×C18C2×C12C18C12C2×C6C2×C4C6C4C22C2
# reps1212442681261824361872

Matrix representation of C2×C108 in GL3(𝔽109) generated by

100
01080
00108
,
10000
0380
0077
G:=sub<GL(3,GF(109))| [1,0,0,0,108,0,0,0,108],[100,0,0,0,38,0,0,0,77] >;

C2×C108 in GAP, Magma, Sage, TeX

C_2\times C_{108}
% in TeX

G:=Group("C2xC108");
// GroupNames label

G:=SmallGroup(216,9);
// by ID

G=gap.SmallGroup(216,9);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,122,118]);
// Polycyclic

G:=Group<a,b|a^2=b^108=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C108 in TeX

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